Mathematical truth has validity independent of place, personality, or human authority. Mathematical relations are not established, nor can they be abrogated, by edict. The multiplication table is international and permanent, not a matter of convention nor of relying upon authority of state or church. The value of π is not amenable to human caprice. The finding of a mathematical theorem may have been a highly romantic episode in the personal life of the discoverer, but it cannot be expected of itself to reveal the race, sex, or temperament of this discoverer. With modern means of widespread communication even mathematical notation tends to be international despite all nationalistic tendencies in the use of words or of type.

In 'Light Thrown on the Nature of Mathematics by Certain Aspects of Its Development', Mathematics in General Education (1940), 256. This is the Report of the Committee on the Function of Mathematics in General Education of the Commission on Secondary School Curriculum, which was established by the Executive Board of the Progressive Education Association in 1932.

Before any great scientific principle receives distinct enunciation by individuals, it dwells more or less clearly in the general scientific mind. The intellectual plateau is already high, and our discoverers are those who, like peaks above the plateau, rise a little above the general level of thought at the time.

Every discoverer of a new truth, or inventor of the method which evolves it, makes a dozen, perhaps fifty, useless combinations, experiments, or trials for one successful one. In the realm of electricity or of mechanics there is no objection to this. But when such rejected failures involve a torture of animals, sometimes fearful in its character, there is a distinct objection to it.

If you want to find out anything from the theoretical physicists about the methods they use, I advise you to stick closely to one principle: don't listen to their words, fix your attention on their deeds. To him who is a discoverer in this field the products of his imagination appear so necessary and natural that he regards them, and would like to have them regarded by others, not as creations of thought but as given realities.

Man is the highest product of his own history. The discoverer finds nothing so grand or tall as himself, nothing so valuable to him. The greatest star is at the small end of the telescope, the star that is looking, not looked after nor looked at.

Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics.

One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser that we are, wiser even than their discoverers, that we get more out of them than was originally put into them.

Our time is distinguished by wonderful achievements in the fields of scientific understanding and the technical application of those insights. Who would not be cheered by this? But let us not forget that human knowledge and skills alone cannot lead humanity to a happy and dignified life. Humanity has every reason to place the proclaimers of high moral standards and values above the discoverers of objective truth.

These creators, makers of the new, can never become obsolete, for in the arts there is no correct answer. The story of discoverers could be told in simple chronological order, since the latest science replaces what went before. But the arts are another storya story of infinite addition. We must find order in the random flexings of the imagination.

To have a railroad, there must have been first the discoverers, who found out the properties of wood and iron, fire and water, and their latent power to carry men over the earth; next the organizers, who put these elements together, surveyed the route, planned the structure, set men to grade the hill, to fill the valley, and pave the road with iron bars; and then the administrators, who after all that is done, procure the engines, engineers, conductors, ticket-distributors, and the rest of the hands; they buy the coal and see it is not wasted, fix the rates of fare, calculate the savings, and distribute the dividends. The discoverers and organizers often fare hard in the world, lean men, ill-clad and suspected, often laughed at, while the administrator is thought the greater man, because he rides over their graves and pays the dividends, where the organizer only called for the assessments, and the discoverer told what men called a dream. What happens in a railroad happens also in a Church, or a State.

Address at the Melodeon, Boston (5 Mar 1848), 'A Discourse occasioned by the Death of John Quincy Adams'. Collected in Discourses of Politics: The Collected Works of Theodore Parker: Part 4 (1863), 139. Note: Ralph Waldo Emerson earlier used the phrase pave the road with iron bars, in Nature (1836), 17.

We have all heard of the puzzle given to Archimedes . His finding that the crown was of gold was a discovery; but he invented the method of determining the density of solids. Indeed, discoverers must generally be inventors; though inventors are not necessarily discoverers.

In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion.
(1987) -- Carl Sagan